This paper introduces the
notion of an intrinsic transversality structure on a Poincaré duality space Xn. Such
a space has an intrinsic transversality structure if the embedding of Xn into its
regular neighborhood Wn+k in Euclidean space can be made “Poincaré transverse”
to a triangulation of Wn+k. This notion relates to earlier work concerning
transversality structures on spherical fibrations, which are known to be essentially
equivalent to topological bundle reductions. Thus, for n ≥ 5, a Poincaré duality
space Xn with a transversality structure on its Spivak normal fibration (i.e.,
with an “extrinsic” transversality structure) is, up to a surgery obstruction,
realizable as a topological manifold. An intrinsic transversality structure,
however, not only guarantees the existence of an extrinsic transversality
structure but gives rise as well to a canonical solution of the resulting surgery
problem. Thus, as our main result, an equivalence is obtained between intrinsic
transversality structures and topological manifold structures. This yields a
number of corollaries, among which the most important is a “local formula
for the total surgery obstruction” which assembles this obstruction to the
existence of a manifold structure on Xn from the local singularities of a
realization of the simple homotopy type of Xn as a (non-manifold) simplicial
complex.
